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The recognition of denatured biological tissue is an indispensable part in the process of high intensity focused ultrasound treatment. As a nonlinear method, multi-scale permutation entropy (MPE) is widely used in the recognition of denatured biological tissue. However, the traditional MPE method neglects the amplitude information when calculating the time series complexity. The disadvantage will affect the recognition effect of denatured tissues. In order to solve the above problems, the method of multi-scale rescaled range permutation entropy (MRRPE) is proposed in this paper. The simulation results show that the MRRPE not only includes the amplitude information of the signal when calculating the signal complexity, but also extracts the extreme volatility characteristics of the signal effectively. The proposed method is applied to the HIFU echo signals during HIFU treatment, and the support vector machine (SVM) is used for recognition. The results show that compared with MPE and the multi-scale weighted permutation entropy (MWPE), the recognition rate of denatured biological tissue based on the MRRPE is higher, up to 96.57%, which can better recognize the non-denatured biological tissues and the denatured biological tissues.

We show that three prominent consumption-based asset pricing models - the Bansal-Yaron, Campbell-Cochrane and Cecchetti-Lam-Mark models - cannot explain the own-history predictability properties of stock market returns. We show this by estimating these models with GMM, deriving ex-ante expected returns from them and then testing whether the difference between realised and expected returns is a martingale difference sequence, which it is not. Furthermore, semi-parametric tests of whether the models' state variables are consistent with the degree of own-history predictability in stock returns suggest that only the Campbell-Cochrane habit variable may be able to explain return predictability, although the evidence on this is mixed.

Uncovering dynamic nonlinearity of slug flow with structural mutation constitutes a challenging problem of significant importance. Rescaled range permutation entropy has been recently introduced as a novel measure for the complexity of chaotic time series with extreme volatility by assigning the ratio of range and standard deviation as the weight for each extracted vector. However, previous work is only limited to the analysis of typical chaotic systems. The intrinsic advancement and the potentiality in the practical application of the algorithm still need to be investigated. Therefore, we firstly conduct simulations on characterizing impulsive signal to demonstrate that rescaled range permutation entropy is more appropriate to characterize the extreme volatility of system. In addition, a practical application in experimental signals analysis of gas–liquid two-phase slug flow is discussed in detail. The result shows that the rescaled range permutation entropy has a significant discernibility for the flow structure sudden change and paves a new way on investigating the dynamic complexity of slug flow.

COVID-19 was first reported in China and quickly spread throughout the world. Weak investor confidence in government efforts to control the pandemic seriously affected global financial markets. This study investigated chaos in China's futures market during COVID-19, focusing on the degree of chaos at different periods during the pandemic. We constructed a phase diagram to observe the attractor trajectory of index futures (IFs). During the COVID-19 outbreak, overall chaos in China's futures market was increasing, and there was a clear correlation between market volatility and the macroenvironment (mainly government regulation). The Hurst index, calculated by rescaled range (R/S) analysis, was 0.46. The price and return of IFs showed long-term correlation and fractal characteristics;the relevant dimensions of the futures market were 2.17. Overall, under the influence of an emergency (COVID-19), chaos in China's financial market intensified, creating a need for timely government intervention and macrocontrol of the market. This study's findings can help improve the government's understanding of the phenomenon of financial chaos caused by emergencies. This study also provides theoretical guidance for controlling financial chaos and maintaining healthy economic development when faced with similar events in the future.

This paper investigates the existence of chaos in concentration dynamics of particulate matter with an aerodynamic diameter less than or equal to $$10\,\upmu \hbox {m}$$ ( $$\hbox {PM}_{10}$$ ) in the greater Athens area (GAA), Greece. It reports findings on three 16-year $$\hbox {PM}_{10}$$ time series recorded by three different air pollution monitoring stations located in GAA and examines if critical fractal epochs with long memory exist. Detrended Fluctuation Analysis (DFA) and Rescaled Range (R/S) Analysis were used via sliding windows of approximately 1-month duration. In all $$\hbox {PM}_{10}$$ time series, several segments were found with critical fractal behaviour and hidden long-memory patterns. All these segments exhibited Hurst exponents above 0.75 and DFA exponents above 1.75. Twelve $$\hbox {PM}_{10}$$ segments with fractality and long memory were commonly identified by both techniques. In one case, long memory was identified concurrently across all three air pollution monitoring stations and in another case, across two stations. The importance of the agreement between two different and independent chaos-analysis techniques is discussed in association with the proper selection of threshold values. This is the second time to address chaos in $$\hbox {PM}_{10}$$ data series in GAA, and the first time to combine two widely accepted techniques, DFA and R/S analysis.

The variation in flow for the Xiaonanhai spring and its main controlling factors were statistically examined using Mann-Kendall trend analysis, rescaled range (R/S) analysis, Morlet wavelet...

When conducting an analysis of nature’s time series, such as meteorological ones, an important matter is a long-range dependence to quantify the global behavior of the series and connect it with other physical characteristics of the region of study. In this paper, we applied the Higuchi fractal dimension and the Hurst exponent (rescaled range) to quantify the relative trend underlying the time series of historical data from 17 of the 34 weather stations located in the Río Bravo-San Juan Basin, Mexico; these data were provided by the National Water Commission CONAGUA) in Mexico. In this way, this work aims to perform a comparative study about the level of persistency obtained by using the Higuchi fractal dimension and Hurst exponent for each station of the basin. The comparison is supported by a climate clustering of the stations, according to the Köppen classification. Results showed a better fitting between the climate of each station and its Higuchi fractal dimension obtained than when using the Hurst exponent. In fact, we found that the more the aridity of the zone the more the persistency of rainfall, according to Higuchi’s values. In turn, we found more relation between the Hurst exponent and the accumulated amount of rainfall. These are relations between the climate and the long-term persistency of rainfall in the basin that could help to better understand and complete the climatological models of the study region. Trends between the fractal exponents used and the accumulated annual rainfall were also analyzed.

Heart acoustics can be used as an early diagnostic tool, to quickly and accurately identify medical patients who may be at risk of unfavourable cardiovascular outcomes. However, at present, there remains no universally accepted standard or technique for detecting abnormalities using Phonocardiograms. To address this, a large database containing normal and abnormal heart sound recordings of patients were studied with the rescaled range technique and the Hurst exponent, a measure of persistence for non-linear dynamic data. Using the Hurst exponent (H) as a benchmark, we compared the values of the Hurst exponent for normal heart sound recordings with that of the abnormal heart sound recordings. For this study, the recording length was limited to the first 10 seconds for all 578 distinct recordings, which were selected randomly from the database. Furthermore, two Hurst exponent values were obtained for each recording, by subjecting them to time intervals of 1 and 2 milliseconds respectively. The results from this study show that heart sound recordings are persistent (H > 0.5) for normal and abnormal heart sound recordings, with the normal recordings being slightly more persistent.

Spatial series based on the histograms of interatomic distances are investigated by the detrended fluctuation analysis and the rescaled range method. Model glycine and alanine peptides are used as the objects of the study. The influence of the monomer type, chain length, and conformation on the values of the Hurst coefficients is analyzed. Most of the spatial series studied are shown to exhibit persistent behavior, that is, to possess long-term memory.

Control Performance Assessment (CPA) has great practical importance. Control quality significantly affects final production throughput, efficiency and environmental impact. There are many approaches starting from time-domain methods, through Minimum Variance, Gaussian and non-Gaussian statistics up to alternative wavelet, fractal or entropy measures. Analysis of production data from process industry shows that signals are often described by non-Gaussian distributions, mostly fat-tail. On the other hand, simulations show that strong disturbances may significantly screen ability of proper detection. This work tests different approaches, i.e. Gaussian standard deviation and fat-tail distribution factors, integral indexes and focuses on persistence measures of rescaled range R/S plot. Robustness of above measures against disturbances with varying statistical properties is investigated. Results confirm that fractal measures may be applied as robust alternative to standard statistics.

This paper aimed to test random walking through the ISX60 market index for the ability to judge market efficiency at a weak level. The study used Serial Correlation Test, the Runs Test, the Variance Ratio Test, as well as the Rescaled Range Test.The population of the study represents of Iraq Stock Exchange. The study concluded accepting the hypothesis of the study that the returns of the ISX60 market index in the Iraqi market for securities does not follow the random walking in general and as a result the Iraq market for securities is inefficient within the weak level of efficiency and the study recommended need a supervisors work in the Iraqi market for securities to activate all means a which will work to communication with information to all investors and thus raise the efficiency of the Iraqi market for securities in order to the avoid of achieving unusual returns by some investors.

Low-permeability reservoirs are characterized by poor physical properties, strong heterogeneity and natural fractures. Accurate identification of the fracture distribution is of great significance for the exploration and development of low-permeability reservoirs. Field outcrop, core and imaging log analyses show that stratabound vertical and high-angle structural fractures are the main types of fractures in the Chang 6 formation in the Huaqing oilfield. The fracture orientations include NE-SW, NNE-SSW, NW-SE, and NNW-SSE. The NE-SE-oriented fractures formed in the Himalayan stage, while the NW-SE fractures formed in the Yanshan period, and the other fractures derived from the fracturing in the Yanshan period during the Himalayan tectonic activity. The traditional rescaled range (R/S) analysis method is upgraded by introducing the finite difference (FD) method in this research, and the newly proposed method is called the R/S-FD method. Core data show that the R/S-FD method can accurately recognize vertical fractures and high-angle fractures. By comparing the fracture development section on the imaging log, it is found that fractures discrimination parameter F has a good linear relationship with both the fracture linear density and the fracture surface density. This method is verified by dynamic data (water injection profiles) and static data (image logging), and the results show that the recognition error is eliminated when the lower limit of F is 0.5, resulting in a crack identification accuracy of 76.9%. Combining the R/S and finite difference methods reduces the artificial recognition error of the traditional R/S method, and the recognition process is automated and intelligent. The R/S-FD method can be used in combination with multiple logging curves. A high fracture recognition accuracy could be accomplished by using this proposed method with only conventional logging curves, of great practical significance to some low-permeability oilfields that lack image logging data.

Anaerobic treatment is a viable alternative for the treatment of agro-industrial waste. Anaerobic digestion reduces organic load and produces volatile fatty acids (VFA), which are precursors of value-added products such as methane-rich biogas, biohydrogen, and biopolymers. Nowadays, there are no low-cost diagnosis and monitoring systems that analyze the dynamic behavior of key variables in real time, representing a significant limitation for its practical implementation. In this work, the feasibility of using the multiscale analysis to diagnose and monitor the key variables in VFA production by anaerobic treatment of raw cheese whey is presented. First, experiments were carried out to evaluate the performance of the proposed methodology under different operating conditions. Then, experimental pH time series were analyzed using rescaled range (R/S) techniques. Time-series analysis shows that the anaerobic VFA production exhibits a multiscale behavior, identifying three characteristic regions (i.e., three values of Hurst exponent). In addition, the dynamic Hurst exponents show satisfactory correlations with the chemical oxygen demand (COD) consumption and VFA production. The multiscale analysis of pH time series is easy to implement and inexpensive. Hence, it could be used as a diagnosis and indirect monitoring system of key variables in the anaerobic treatment of raw cheese whey.

Automatic analysis of electrocardiographic (ECG) signals, including the heart rate variability (HRV), makes it possible to assess the health status of patients, by reducing the likelihood of human error and providing an optimal and relatively accurate result. HRV is an important information indicator for diagnosing and predicting cardiovascular disease, which is based on measuring the intervals between heartbeats (known as RR-time intervals) derived from ECG signals. HRV has been used for research in various scientific fields, including information technology where it is used to create software products for automatic analysis of ECG signals in order to gain additional knowledge about the behavior of RR fluctuations in normal and disease states. ECG signals are non-stationary and one of the most suitable methods for analysis are the fractal methods. This article presents the results of the study of the fractal and multifractal properties of real ECG signals, combined into two groups: ECG signals of healthy subjects as well as patients with cardiovascular disease (arrhythmia), using the methods: Rescaled range (R/S) analysis and Multifractal Detrended Fluctuation Analysis (MFDFA). The obtained values ​​of the studied parameters are used to distinguish healthy subjects from sick ones, by applying statistical analysis. The statistical analysis is performed by applying a t-test to determine the statistical significance of the studied ECG signals and Receiver Operating Characteristic (ROC) analysis to assess the quality of the selected methods. The obtained results show that the fractal methods used are suitable for analysis of the dynamics of RR intervals and for distinguishing the healthy subjects from those with pathological diseases.

The contribution of this article is mainly to develop a new stochastic sequence forecasting model, which is also called the difference iterative forecasting model based on the Generalized Cauchy (GC) process. The GC process is a Long-Range Dependent (LRD) process described by two independent parameters: Hurst parameter H and fractal dimension D. Compared with the fractional Brownian motion (fBm) with a linear relationship between H and D, the GC process can more flexibly describe various LRD processes. Before building the forecasting model, this article demonstrates the GC process using H and D to describe the LRD and fractal properties of stochastic sequences, respectively. The GC process is taken as the diffusion term to establish a differential iterative forecasting model, where the incremental distribution of the GC process is obtained by statistics. The parameters of the forecasting model are estimated by the box dimension, the rescaled range, and the maximum likelihood methods. Finally, a real wind speed data set is used to verify the performance of the GC difference iterative forecasting model.

The paper discusses a technique for detecting cyberattacks on computer networks, based on identifying anomalies in network traffic by assessing its self-similarity and determining the impact of cyber attacks using statistical methods. The proposed technique includes three stages, at which the analysis of the self-similarity property for the reference traffic is performed (using the methods of the Dickey-Fuller test, rescaled range, and detrended fluctuation), the analysis of the self-similarity property for the real traffic (by the same methods) and additional processing of time series with statistical methods (methods of moving average, Z-Score, and CUSUM). The issues of software implementation of the proposed approach and the formation of a dataset containing network packets are considered. The experimental results demonstrated the presence of self-similarity in network traffic and confirmed the high efficiency of the proposed method. This technique allows detecting cyberattacks in real or near real time.

We have identified the lithologies from well logs and available core data through the cluster and neural network analysis in the Krishna-Godavari (KG) basin. The unsupervised hierarchical cluster analysis (HCA) has been used to find out the dissimilarity behaviour of pairwise well log data associated with each lithological unit as a measure of proximity. Whereas, supervised neural network analysis has been used for the identification of lithologies of un-cored portion of well using the wireline logs and associated lithologies identified from cores in the same or nearby well. Initially, the persistence behaviour of the wireline logs is confirmed by rescaled range (R/S) analysis. These log data follow a local trend of lithological variation and hence can be used for lithology identification. Subsequently, we have used HCA and multi-layer feed forward (MLF) neural network to envisage the lithological sequence of varying thickness (thick beds of the order of 6 m; thin beds up to 0.3048 m) through the analysis of gamma-ray, bulk density (RHOB), neutron porosity (Φ), sonic transit time (Δt) and photoelectric factor downhole logs. The results using the log data from the Expedition 02 of Indian National Gas Hydrates Program (NGHP-Exp.-02) demonstrate that these non-traditional approaches are suitable for analysing formation lithologies where core data associated with discriminating finer beds are available. The HCA and MLF network-predicted lithologies, made in this study, match realistically with the core derived lithologies, demonstrating their efficacy. Thus, the approach is quite useful for providing quick and accurate information on subsurface lithologies.

Ronnie Araneda gratefully acknowledges financial support from the Spanish Regional Government of Galicia (Xunta de Galicia) and the European Union through the Predoctoral Grant reference ED481A-2018/162. Maria Bermudez was supported by the European Union H2020 research and innovation program under the Marie Skodowska-Curie Grant Agreement No. 754446 and the Research and Transfer Fund of the University of Granada-Athenea3i. The authors would like to thank Manuel Alvarez and Victor Penas for their comments and support during the development of the study., Study region: Mozambique. Study focus: Mozambique does not currently have the necessary tools for systematic monitoring and forecasting of drought at a subnational scale. The purpose of this study was to characterize drought conditions and trends throughout the country and to evaluate the influence of major climatic drivers on drought events (period 1950–2019). Drought conditions were studied by means of the Standardized Precipitation and Evapotranspiration Index (SPEI) and run theory. The principal component analysis technique and the k-means clustering method were applied for defining homogenous drought regions. The Mann-Kendall trend test and Rescaled Range statistical analysis were used for defining the temporal characteristics of drought. The cross-correlation method, a spectral analysis based on the Fast Fourier Transform and a cross-wavelet analysis, were used to identify possible climate drivers. The results are ultimately intended to contribute to the development of a drought monitoring system in this country. New hydrological insights for the region: Three homogeneous drought regions can be defined in Mozambique. The South and Centre regions showed more intense and severe drought events. In all regions, a significant trend towards a higher incidence of droughts and long-term a persistence were found. El Niño-Southern Oscillation and Darwin Sea Level Pressure anomalies were identified as significant drivers of drought variability, especially in the southern regions. These climate indices can be used as predictors in drought forecasting models., Spanish Regional Government of Galicia (Xunta de Galicia), European Commission ED481A-2018/162, European Union H2020 research and innovation program under the Marie Skodowska-Curie Grant 754446, Research and Transfer Fund of the University of Granada-Athenea3i

The characterization of time series requires knowledge of certain para- meters. One of those parameters is the Hurst exponent, which is an indicator of long-range dependence characteristics. Rescaled range (R/S), proposed by E. Hurst, is the most commonly used method to compute this exponent. On the other hand, wavelet analysis is known to reflect better the nonlinear dynamics of the biological, climatic, or economic series than the statistical tools often used for this analysis. The average wavelet coefficient (AWC) is a wavelet method that has been used for the last years to compute the Hurst exponent. In this paper, we present a modification to the AWC method, and we compare its performance with the original version of AWC and with R/S methods. The results obtained for the synthetic series were so promising that we decided to apply our proposal in rainfall series. Therefore, these results were compared with the ones reported from La Pampa. After that, series from different climatic regions of the Argentine Republic were analyzed.

The large-conductance voltage- and Ca2+-activated K+ channels (BK) are encoded in humans by the Kcnma1 gene. Nevertheless, BK channel isoforms in different locations can exhibit functional heterogeneity mainly due to the alternative splicing during the Kcnma1 gene transcription. Here, we would like to examine the existence of dynamic diversity of BK channels from the inner mitochondrial and cellular membrane from human glioblastoma (U-87 MG). Not only the standard characteristics of the spontaneous switching between the functional states of the channel is discussed, but we put a special emphasis on the presence and strength of correlations within the signal describing the single-channel activity. The considered short- and long-range memory effects are here analyzed as they can be interpreted in terms of the complexity of the switching mechanism between stable conformational states of the channel. We calculate the dependencies of mean dwell-times of (conducting/non-conducting) states on the duration of the previous state, Hurst exponents by the rescaled range R/S method and detrended fluctuation analysis (DFA), and use the multifractal extension of the DFA (MFDFA) for the series describing single-channel activity. The obtained results unraveled statistically significant diversity in gating machinery between the mitochondrial and cellular BK channels.

The objective of this work is to generate an effective selection of cased-hole completions that contributes to increase the production of M field, contemplating all the necessary technical criteria to ensure that the evaluation formation and static fractal analysis could be advantageous for the reservoir characterization and the determination of well productivity parameters. We used the Statistic-Fractal analysis inside of petrophysical model known as fractal and geostatistical models whose main purpose is to analyze data for reservoir characterization. That is, through separating data from production log tools and resistivity logs, it is possible to have an accurate picture of the hydrocarbon path through the identification of the repeatable marks of data along the selected well logs from cased-hole completions of the field in study. That representative mark, associated with the knowledge of the internal characteristics of the reservoir (porosity, permeability, fracture systems and faults, Water-Oil Contact) help to obtain better results. For this scrutiny we analyzed 12 wells (1, 11, 51, 62, 61, 3A, 3Are, 21, 21are, 31,32, 64), from M field in Southeastern Mexico, deep onshore carbonates (average reservoir depth is 6,493 m) producing light oil, with different completions, which include slotted liner, cemented liner with selective perforations and open-hole completions. This paper describes the process of analysis and selection of new potential intervals to complete and explains the workflow used during the design stage. Statistic-fractal analysis in petrophysical model techniques are discussed together with lessons learned from the wells in terms of production behavior. The importance of this work lies in improving the selection of new locations and completions in mature fields.

This article is devoted to the comparative analysis of four statistical methods: Variance-time, Rescaled Range (R/S), Periodogram and Detrended Fluctuation Analysis (DFA) to study the fractal time series that are simulated by applying an algorithm, based on the Fractional Gaussian Noise (FGN). The evaluation of the statistical methods studied was performed with respect to the precision parameter in determining the Hurst exponents of the simulated fractal time series, with DFA being determined as the most accurate method. The statistical methods studied are characterized by universal against the data examined. In the article, the DFA method is applied to the analysis of two independent types of real-time data representing continuous time series: in the field of cardiology in the study of heart intervals (RR time series) obtained from digital electrocardiographic (ECG) signals and in the field of the energy for the analysis of time series data for the hourly day-ahead spot prices of the Bulgarian electricity market for the period 2016-2019. The results of the application of fractal analysis in modern information technologies related to fractal theory presented in the article reveal great potential and new perspectives in modelling, processing and analysis of processes and signals in various scientific and technical fields.

1Summary1.1BackgroundThe outbreak of the new coronavirus infection broke out in Wuhan City, Hubei Province in December 2019, and has spread to 97 countries and regions around the world. Apart from China, there are currently three other severely affected areas, namely Italy, South Korea, and Iran. This poses a huge threat to China’s and even global public health security, challenges scientific research work such as disease surveillance and tracking, clinical treatment, and vaccine development, and it also brings huge uncertainty to the global economy. As of March 11, 2020, the epidemic situation in China is nearing its end, but the epidemic situation abroad is in the outbreak period. Italy has even taken measures to close the city nationwide, with a total of 118,020 cases of infection worldwide.1.2MethodThis article selects the data of newly confirmed cases of COVID-19 at home and abroad as the data sample. Among them: the data of newly confirmed cases abroad is represented by Italy, and the span is from February 13 to March 10. The data of newly confirmed cases at home are divided into two parts: Hubei Province and other provinces except Hubei Province, spanning from January 23 to March 3, and with February 12 as the cutting point, it”s divided into two periods, the growth period and the recession period. The rescaled range (R / S) analysis method and the dimensionless fractal Hurst exponent are used to measure the correlation of time series to determine whether the time series conforms to the fractal Brownian motion, that is, a biased random process. Contrast analysis of the meaning of H value in different stages and different overall H values in the same stage.1.3ResultsBased on R / S analysis and calculated Hurst value of newly confirmed cases in Hubei and non-Hubei provinces, it was found that the H value of Hubei Province in the first stage was 0.574, which is greater than 0.5, indicating that the future time series has a positive correlation and Fractal characteristics; The H value in the second stage is 1.368, which is greater than 1, which indicates that the future epidemic situation is completely preventable and controllable, and the second stage has a downward trend characteristic, which indicates that there is a high probability that the future time series will decline. The H values of the first and second stages of non-Hubei Province are 0.223 and 0.387, respectively, which are both less than 0.5, indicating that the time series of confirmed cases in the future is likely to return to historical points, and the H value in the second stage is greater than that in the first stage, indicating that the time series of confirmed cases in the second stage is more long-term memory than the time series of confirmed cases in the first stage. The daily absolute number of newly confirmed cases in Italy was converted to the daily growth rate of confirmed cases to eliminate the volatility of the data. The H value was 1.853, which was greater than 1, indicating that the time series of future confirmed cases is similar to the trend of historical changes. The daily rate of change in cases will continue to rise.1.4ConclusionAccording to the different interpretation of the H value obtained by the R / S analysis method, hierarchical isolation measures are adopted accordingly. When the H value is greater than 0.5, it indicates that the development of the epidemic situation in the area has more long-term memory, that is, when the number of confirmed cases in the past increases rapidly, the probability of the time series of confirmed cases in the future will continue the historical trend. Therefore, it is necessary to formulate strict anti-epidemic measures in accordance with the actual conditions of various countries, to detect, isolate, and treat early to reduce the base of infectious agents.

In this paper, to solve the problem of detecting network anomalies, a method of forming a set of informative features formalizing the normal and anomalous behavior of the system on the basis of evaluating the Hurst (H) parameter of the network traffic has been proposed. Criteria to detect and prevent various types of network anomalies using the Three Sigma Rule and Hurst parameter have been defined. A rescaled range (RS) method to evaluate the Hurst parameter has been chosen. The practical value of the proposed method is conditioned by a set of the following factors: low time spent on calculations, short time required for monitoring, the possibility of self-training, as well as the possibility of observing a wide range of traffic types. For new DPI (Deep Packet Inspection) system implementation, algorithms for analyzing and captured traffic with protocol detection and determining statistical load parameters have been developed. In addition, algorithms that are responsible for flow regulation to ensure the QoS (Quality of Services) based on the conducted static analysis of flows and the proposed method of detection of anomalies using the parameter Hurst have been developed. We compared the proposed software DPI system with the existing SolarWinds Deep Packet Inspection for the possibility of network traffic anomaly detection and prevention. The created software components of the proposed DPI system increase the efficiency of using standard intrusion detection and prevention systems by identifying and taking into account new non-standard factors and dependencies. The use of the developed system in the IoT communication infrastructure will increase the level of information security and significantly reduce the risks of its loss.

Hydrocarbon (HC) and non-hydrocarbon (NHC) zones have been characterized using fractal analysis of wireline log data. Well-log responses such as spontaneous potential (SP), gamma ray (GR), resistivity (LLD, LLS, MSFL), neutron porosity (NPHI), bulk density (RHOB) log have been analyzed from five boreholes of Bhogpara oil field using rescaled range (R/S) and power spectrum density (PSD) techniques. Primarily, both the techniques were applied on the log responses associated with the traditionally identified potential and non-potential zones of 100 m thickness. The Hurst exponent (HPSD) obtained using PSD analysis for GR, NPHI and RHOB log, shows higher values for HC zones as compared to the NHC zones. Furthermore, to understand the scaling pattern more explicitly for main potential sector of the reservoir. Both of the techniques were applied to the wireline log data that are associated with traditionally defined HC and NHC zones of different thickness. The Hurst exponent (HR/S) obtained using R/S analysis was found to be of relatively higher values for HC zones as compared to the NHC zones for all wireline log response. Again opposite trend was also observed for HPSD values that obtained from GR, MSFL, NPHI and RHOB log responses only. Hence, the HC and NHC zones can be delineated from fractal analysis of wireline logs and also overcome the chance of misinterpretation, which is quite possible in the case of hydrocarbon zone detection in traditional reservoir characterization.

This article introduces an integrated photoplethysmographic (PPG) based cardiovascular monitoring system that consists of an individually portable PPG device for recording photoplethysmographic signals and a software system with a serverless architecture for processing, storing, and analyzing the obtained signals. The portable device uses the optical plethysmography technique for measuring blood volume in blood vessels. The device was tested and validated by a comparative analysis of three photoplethysmographic signals and one Electrocardiographic signal registered simultaneously in the target subject. The comparative analysis of these signals shows insignificant deviations in the obtained results, with the mean squared error between the studied signals being less than 21 ms. This deviation cannot affect the results that were obtained from the analysis of the interval series tested. Based on this result, we assume that the detected signals with the proposed device are realistic. The designed software system processes the registered data, performs preprocessing, determines the pulse rate variability, and performs mathematical analysis of PP intervals. Two groups of subjects were studied: 42 patients with arrhythmia and 40 healthy controls. Mathematical methods for data analysis in time and frequency domain and nonlinear methods (Poincar&eacute, plots, Rescaled Range Plot, Detrended Fluctuation Analysis, and MultiFractal Detrended Fluctuation Analysis) are applied. The obtained results are presented in tabular form and some of them in graphical form. The parameters studied in the time and frequency domain, as well as with the nonlinear methods, have statistical significance (p &lt, 0.05) and they can distinguish between the two studied groups. Visual analysis of PP intervals, based on Poincare&rsquo, s nonlinear method, provides important information on the physiological status of patients, allowing for one to see at a glance the entire PP interval series and quickly detect cardiovascular disorders, if any. The photoplethysmographic data of healthy individuals and patients diagnosed with arrhythmia were recorded, processed, and examined through the system under the guidance of a cardiologist. The results were analyzed and it was concluded that this system could serve to monitor patients with cardiovascular diseases and, when the condition worsens, a signal could be generated and sent to the hospital for undertaking immediate measures to stabilize patient&rsquo, s health.

The long-range correlations within the current signal time series of the Beta vulgaris vacuolar membrane under the influence of organolead compound (Met3PbCl) are investigated. The current time series is transformed into a dwell time series. Then the rescaled range and detrended fluctuations analyses are used. It is shown that the presence of Met3PbCl in the solution decreases the mean value of the Hurst exponent and therefore influences the long-range correlations in ionic channel current. This observation is statistically significant. An ion channel model is built and the experimental results reconstructed and analysed.

This paper investigates the complexity of soil radon time series (RTS) recorded from March 2017 to April 2018 in Pakistan via RTM 1688-2 monitor, in direct linkage to parallel series of temperature, relative humidity, and pressure. Several complexity measures are employed. The normalized Lempel Ziv Complexity (LZC) of the complete RTS is 0.30 and 0.14, 0.22, and 0.14 for the complete time series of temperature, relative humidity and pressure respectively. LZC ranges between 0.3 and 0.5 for RTS and between 0.1 and 0.3 for temperature (T), relative humidity (RH) and pressure (P). Hurst exponent (HE) values via rescaled range (R/S) analysis show an inverse relationship with LZC. HE ranges between 0.60 ≤ HE ≤ 0.98 for RTS, 0.60 ≤ HE ≤ 0.92 for T series, 0.61 ≤ HE ≤ 0.93 for RH series, and 0.87 ≤ HE ≤ 0.98 for P series. Permutation Entropy (PE) is 1.7809 for complete RTS with a minimum of 1.7668 in August 2018. Highest value of Sample Entropy (SE) is 1.3716 for February 2018. The highest symbolic Normalized Corrected Shannon Entropy (NCSE) value is found in January 2018 with a value of 0.7864. Finally, the Fractal Dimensions varied significantly for calculation with the Sevcik's, Higuchi's and Katz's methods. Common dates of below threshold areas are reported. The different aspects of the derived results are discussed.

Long range memory in share indices show temporal dependence between observations spaced by long intervals of time and has distinct non-periodic cycles. This paper examines the presence of long memory of various indices of National Stock Exchange (NSE). The data consists of closing values of indices over different periods of time. The tests applied to examine long memory are Hurst exponent, Manderbolt-Hurst exponent, Lo’s rescaled-range analysis and Geweke and Porter-Hudak (GPH) test. The results of the estimated Hurst exponent, Manderbolt-Hurst exponent and GPH test show that invariably all NSE indices series have long memory. However, the results of Lo’s rescaled-range analysis indicate the absence of long memory for all indices. Key words: Long memory, rescaled range analysis, fractional dimension, Hurst exponent, GPH test, NSE indices

A fractal analysis from rainfall events registered in a semiarid region was carried out. The analysis was executed for Baja California, Mexico, a region that presents a high climatological variability. Rainfall data from 92 climate stations distributed along the region of study with at least 30 years of records were used. By studying the rainfall series patterns, the Hurst exponent values were obtained (both spatial and temporal) as well as their relation with the variables for annual average temperature, annual average rainfall, altitude and climatological distribution. The rescaled range method, box counting method, and multifractal detrended fluctuation analysis were successfully applied, having as a result the average value of the Hurst exponent for different time scales. Data showed that the daily rainfall series tend to present a persistent pattern; besides, the Hurst exponent values ( H u ) were related to the type of climate, altitude, rainfall regime, and temperature of the studied area. The analysis of the Hurst exponent for different time scales showed that, at a smaller time scale, the Hurst value tends to increase; thus, the series present a stronger persistent behavior. Moreover, it can be confirmed that the fractal theory methodology allows analyzing the behaivor of a climate variable, in this case rainfall.

In recent years, global warming has attracted great attention around the world. Temperature change is not only involved in global climate change but also closely linked to economic development, the ecological environment, and agricultural production. In this study, based on temperature data recorded by 756 meteorological stations in China during 1951–2013, the spatial and temporal stability characteristics of annual temperature in China and its first-level basins were investigated using the rank correlation coefficient method, the relative difference method, rescaled range (R/S) analysis, and wavelet transforms. The results showed that during 1951–2013, the spatial variation of annual temperature belonged to moderate variability in the national level. Among the first-level basins, the largest variation coefficient was 114% in the Songhuajiang basin and the smallest variation coefficient was 10% in the Huaihe basin. During 1951–2013, the spatial distribution pattern of annual temperature presented extremely strong spatial and temporal stability characteristics in the national level. The variation range of Spearman’s rank correlation coefficient was 0.97–0.99, and the spatial distribution pattern of annual temperature showed an increasing trend. In the national level, the Liaohe basin, the rivers in the southwestern region, the Haihe basin, the Yellow River basin, the Yangtze River basin, the Huaihe basin, the rivers in the southeastern region, and the Pearl River basin all had representative meteorological stations for annual temperature. In the Songhuajiang basin and the rivers in the northwestern region, there was no representative meteorological station. R/S analysis, the Mann-Kendall test, and the Morlet wavelet analysis of annual temperature showed that the best representative meteorological station could reflect the variation trend and the main periodic changes of annual temperature in the region. Therefore, strong temporal stability characteristics exist for annual temperature in China and its first-level basins. It was therefore feasible to estimate the annual average temperature by the annual temperature recorded by the representative meteorological station in the region. Moreover, it was of great significance to assess average temperature changes quickly and forecast future change tendencies in the region.

The Hurst phenomenon is a well-known feature of long-range persistence first observed in hydrological and geophysical time series by E. Hurst in the 1950s. It has also been found in several cases in turbulence time series measured in the wind tunnel, the atmosphere, and in rivers. Here, we conduct a systematic investigation of the value of the Hurst coefficient H in atmospheric surface-layer data, and its impact on the estimation of random errors. We show that usually $$H > 0.5$$ , which implies the non-existence (in the statistical sense) of the integral time scale. Since the integral time scale is present in the Lumley–Panofsky equation for the estimation of random errors, this has important practical consequences. We estimated H in two principal ways: (1) with an extension of the recently proposed filtering method to estimate the random error ( $$H_p$$ ), and (2) with the classical rescaled range introduced by Hurst ( $$H_R$$ ). Other estimators were tried but were found less able to capture the statistical behaviour of the large scales of turbulence. Using data from three micrometeorological campaigns we found that both first- and second-order turbulence statistics display the Hurst phenomenon. Usually, $$H_R$$ is larger than $$H_p$$ for the same dataset, raising the question that one, or even both, of these estimators, may be biased. For the relative error, we found that the errors estimated with the approach adopted by us, that we call the relaxed filtering method, and that takes into account the occurrence of the Hurst phenomenon, are larger than both the filtering method and the classical Lumley–Panofsky estimates. Finally, we found that there is no apparent relationship between H and the Obukhov stability parameter. The relative errors, however, do show stability dependence, particularly in the case of the error of the kinematic momentum flux in unstable conditions, and that of the kinematic sensible heat flux in stable conditions.

Global Navigation Satellite System (GNSS) is an efficient space geodetic method for precise spatial coordinates determination and time series generation. Time series analysis is very important in in order to find tendencies and make better predictions. In this study we, analyze time series of coordinates ENU (East, North, Up) to determine its performance of GNSS network in the Ecuador using Hurst coefficient estimation. This network is composed by 30 stations. Hurst analysis was done using two approximations. One is Rescaled Range (R/S) technic, and the second is Wavelet transformation. Times series of ENU coordinates were attained from the GNSS stations diary solutions to calculate of Hurst coefficient. Hurst coefficient estimated using both methods were higher than 0.5 which is an indication that the time series indicate persistent properties, that is, a positive correlation.

A monthly average solar green coronal index time series for the period from January 1939 to December 2008 collected from NOAA (The National Oceanic and Atmospheric Administration) has been analysed in this paper in perspective of scaling analysis and modelling. Smoothed and de-noising have been done using suitable mother wavelet as a pre-requisite. The Finite Variance Scaling Method (FVSM), Higuchi method, rescaled range (R/S) and a generalized method have been applied to calculate the scaling exponents and fractal dimensions of the time series. Autocorrelation function (ACF) is used to find autoregressive (AR) process and Partial autocorrelation function (PACF) has been used to get the order of AR model. Finally a best fit model has been proposed using Yule-Walker Method with supporting results of goodness of fit and wavelet spectrum. The results reveal an anti-persistent, Short Range Dependent (SRD), self-similar property with signatures of non-causality, non-stationarity and nonlinearity in the data series. The model shows the best fit to the data under observation.